I too found it really surprising that the reverse-time equation has a simple closed form. Like, surely breaking a glass is easier than unbreaking it? That’s part of what got me interested in this stuff in the first place!

If you haven’t seen it yet, highly recommend the blogs of Sander Dieleman & Yang Song (who co-invented the SDE interpretation).

Thanks for reading. The 2D simulation section might be more interesting on a first read — it makes the math less mysterious, I hope!

Theta represents all the model params — all the weights in the neural network. The convention is to write theta for the “learned” score function and omit theta for the “true” score function.

I personally find the SDEs the most intuitive, and the deterministic ODE / consistency models / rectified flow stuff as ideas that are easier to understand after the SDEs. But not everyone agrees!

In the “Sad Irons” chapter of Caro’s LBJ biography, he talks about the pre-electrification lives of Texas farmers. In comparison with that, our whole day is leisure.

Similarly: As late as 1900, the poor in Europe were so severely malnourished that growth stunting was common. Look at Our World in Data’s charts of height over time. Or Robert Fogel’s “The Escape from Hunger and Premature Death.”

Etc. Etc.

credit bubbles often pop when some institution can't cover its obligations, but asset bubbles don't seem to need such a failure — and can deflate on their own.

Other references I found interesting:

  - Cover and Thomas's "Elements of Information Theory" shows some interesting connections between Kelly betting and optimal message encoding.
  - Ed Thorp, the inventor of card counting, has a nice compendium of papers on this in "The Kelly Capital Growth Investment Criterion".

Comments URL: https://news.ycombinator.com/item?id=23894425

Points: 4

# Comments: 1

E.g., say that the random numbers are uniformly distributed between -200 and 200. If somebody says a number like 425, then I know they weigh at least 225 pounds. And the probability they weigh more than 225 - k pounds is 1 - k/400.

* "hard landing" makes the airline industry seem tumultuous and exciting

* "a man for all markets" is the autobiography of ed thorpe, father of card counting and quant hedge funds

* "unix: a history and a memoir" is a mischievous first hand account from brian kernighan (of unix / c / awk fame)

I wouldn't say that you can't do this stuff without category theory, but it does make it easier / clearer. (Similarly, you can do a lot of geometry without coordinates, but coordinates definitely make some stuff easier / clearer.)

[1] https://www.wikiwand.com/en/Brouwer_fixed-point_theorem#/A_p... [2] https://www.wikiwand.com/en/Eilenberg%E2%80%93MacLane_space

Call a real number "algebraic" if it's a zero to some polynomial with rational coefficients. (e.g. \sqrt{2} is algebraic since it's a zero for x^2 - 2). Claim: There exist non-algebraic ("transcendental") numbers. Proof: There are only countably many polynomials, and so there are only countably many algebraic numbers, but there are uncountably many reals. Similarly, there are numbers that aren't Turning-computable. Etc.

Comments URL: https://news.ycombinator.com/item?id=19565364

Points: 2

# Comments: 0

But: 42% is the effective tax rate (not marginal, average). And this is pretty close to 45%. For a hypothetical person making $500k or more as regular income in CA (as many of the posted salaries above would be), they would indeed pay about $210k in taxes plus Medicare plus Social Security.

For a $500k income in CA

    29.59% Federal Income Tax
  + 10.48% CA Income Tax
  +  1.59% Social Security
  +  1.99% Medicate
  ----------------------------
  = 43.65% total tax rate.

> Both the historical and the control bucket used version A of the website, and they are consistent in their 2.0% conversion rate. Version B is different, and it appears to have a different conversion rate of 2.5%. So why should it not have a future conversion rate close to 2.5%?

It's all a matter of degree. You'd model B's rate as closer to 2.5%, but probably not centered around 2.5%. As you observe more data, the prior becomes less important. E.g., with 10k samples as in the original example, if you used Beta(2+1,100-2+1) as your prior, your posterior would be Beta(252+1, 10100-2+1) as your posterior, which is centered at 2.495%. But if you only had 1000 samples (and 25 conversions), you'd get a distro centered at 2.45%. And if you only had 200 samples (and 5 conversions), you'd get a distro centered at 2.33%. Etc.

> Let's replace the website with a 6-sided die. Historically, the probability of throwing a 3 was 1/6. Now you replace your die with a different die and throw it 10,000 times; the 3 comes up 2560 times. If I had to guess how many times the 3 comes up the next 10,000 throws, I certainly would bet that it's closer to 2560 times than to 1667 times.

In the case of a die where you believe any weighting of the faces is equally likely, this would be true. So this may be an appropriate model in this case. But in the case of the website, I don't think the conversion rates are equally likely, even for a new, un-tested site. If the historical conversion rate is 2.0%, and I'm forced to bet on the most likely conversion for a new (never before seen) variant B, I'd much rather bet on a number near 2.0% than a number like 99%.

> Case B: The historical version A of the online shop did not have any influence on the conversion rate during the testing of version B (compare the dice example above). Then both ranges are equally plausible.

This is exactly what I'm claiming is not true. It's not that A influences B, it's that A tells you something about the likely range of A and B (in this specific case of an e-commerce site). (The reason I chose the ranges [2.0%, 2.5%] vs [2.5%, 3.0%] is that if you model B independently, you'd be indifferent between these ranges; but if you use A to inform a prior, you'd prefer [2.0%, 2.5%].)